Amelioration of Verdegay s Approach for Fuzzy Linear Programs with Stochastic Parameters

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1 ` Iranan Journal of Manageent Stude (IJMS) Vol. 11, No. 1, Wnter 2018 Prnt ISSN: pp Onlne ISSN: DOI: / Aeloraton of Verdegay Approah for Fuzzy Lnear Progra wth Stohat Paraeter Seyed Had Naer, Sal Bavand Departent of Matheat, Unverty of Mazandaran, Babolar, Iran (Reeved: June 21, 2017; Reved: Deeber 11, 2017; Aepted: Deeber 26, 2017) Abtrat Th artle exane a new approah whh olve Lnear Prograng (LP) proble wth tohat paraeter a a generalzed odel of the fuzzy atheatal odel analyzed by Verdegay. An expetaton odel provded for olvng the proble. A ult-paraetr prograng appled to ae to a oluton wth dfferent dered degree a well a proble ontrant. Addtonally, we preent a nueral exaple to deontrate the tate and ethod effeny. Keyword Fuzzy deon akng, Verdegay' ethod, tohat lnear prograng, expetaton ethod, ult-paraetr prograng. Correpondng Author, Eal: albavand@yahoo.o

2 72 (IJMS) Vol. 11, No. 1, Wnter 2018 Introduton Fuzzy Lnear Prograng (FLP) progra wll appear when a knd of abguty ourred n the paraeter of the odel and/or n the lak of deternt nforaton n the laal lnear prograng (LP) progra. In th ae, the entoned odel wll be etablhed baed on a type of abguty ntead of the rp data uh a fuzzy nput and o on. In the lterature of fuzzy optzaton, there are a lot of reearhe whh are foued on fuzzy lnear prograng. An nteretng approah ung paraetr prograng ethod n the fuzzy lnear prograng. Afterward, varou type of LP proble along wth ther olvng ethod have been preented by dfferent author. Alo, th portant to dtnguh between flexblty and unertanty whh appear n the paraeter of odel nludng the a oeffent, the oeffent atrx, and t ontrant and alo n the rght-hand-de data. Flexblty onept odeled ung fuzzne and reflet the onept that feablty of the oluton for every ontrant wll be vald baed on t atfaton degree. Furtherore, every ontrant ha an aepted tolerane whh predeterned by the deon aker. Moreover, there unertanty that onern an obetve varablty n the dered odel paraeter (rando unertanty) or the lak of oe nforaton of the paraeter value. Verdegay (1982) howed that an LP proble wth the rp target and oe fuzzy ontrant equal to a oon paraetr LP odel and thu, we are able to ue paraetr approahe to olve thee FLP proble. Cadena and Verdegay (2000) provded a ultobetve LP proble, where the obetve oeffent n the obetve funton appeared a fuzzy nuber, and alo appled fuzzy orderng approah to olve thee odel. Naer and Bavand (2017) ondered a Stohat Interval-Valued Lnear Fratonal Prograng (SIVLFP) proble, where n ther odel, the oeffent and alar of the obetve funton are fratonal nterval, and tehnologal oeffent and the quantte of the ontrant n the entoned odel were rando varable wth the

3 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 73 pef dtrbuton. Alo, an nteratve approah wa propoed for Mult-Obetve Fuzzy Stohat Lnear Prograng (MOFSLP) progra by Mohan and Nguyen (2001). After that, Ikander (2003) eployed a tate of fuzzy weghted obetve funton to olve a MOFSLP proble. Alo after that, other knd of LP wth fuzzy rando oeffent tuded by oe author (Ikander, 200a; Ikander, 200b; Ikander, 2005). Furtherore, Stohat Lnear Prograng (SLP) tuded by Ben Abdelazz and Mar (2005), n whh they ued fuzzy and/or rp nequalty for every ontrant ntead of the probablty dtrbuton. In fat, they ued the α ut approah for defuzzfng the aoated probablty dtrbuton. Naer et al. (2005) ondered a FLP proble and uggeted plex ethod to olve thee progra ung lnear rankng funton. In the urrent deade, alo oe erou tude are nvetgated n the lterature. One of thee work gven by Luhandula (2006). In th work, the author preented a urvey of the eental odel and ethod whh are preented on fuzzy tohat prograng area. For Goal Prograng wth Fuzzy Stohat paraeter (FSGP) proble, where all paraeter are ondered a knd of fuzzy rando varable, Hop (Hop, 2007a; Hop, 2007b; Hop, 2007) ntrodued a novel approah for olvng thee proble. Roelfanger (2007) tuded an LP wth ult-rtera rp, fuzzy or alo tohat value. Reently, Attar and Naer (201) ntrodued a novel onept for the Fuzzy Matheatal Prograng (FMP) whh onern the feablty of the optal oluton of thee odel a an extenon of the laal onept of feablty, alo t ha appeared n the lterature of operaton reearh. They ut onder fuzzne n the ontrant of the entoned odel whle n the any real tuaton a knd of abguty ourred n the obetve oeffent. So, we are gong to extend ther odel to a generalzed for, whh the obetve oeffent are nludng tohat paraeter. Baed on Verdegay ethod whh appled for fuzzy lnear prograng, we ugget an aeloraton ethod to olve FLP progra whh nlude tohat paraeter n the obetve funton. Dfferent part of th reearh are prepared n fve eton. In

4 7 (IJMS) Vol. 11, No. 1, Wnter 2018 Seton 2, we preent the eental onept and reult on tohat prograng that wll be ueful n our ethodology. In Seton 3, we defne two new onept: feable and effent oluton, and then baed on the theoretal duon, we provde a oputatonal ethod for obtanng a oluton for the entoned proble. In Seton, an applaton of the ethod derbed n FLP proble. We fnally preent oe portant reult n the lat eton. Stohat Prograng The aor part of th eton regardng the fundaental and neeary onept and defnton of probablty theory taken fro Caella and Berger (2001) and Grett and Strzaker (2001). Hene, to brng the prelnare, we ot the detal of the bakground here. In partular, the Stohat Lnear Prograng (SLP) proble are onentrated. Furtherore, we ntrodue a new deon akng odel aordng to E-odel, whh one of the ot effent odel n the ene of the SLP progra. Aue the followng Stohat Lnear Prograng (SLP) progra: Max z ( x ) x n t.. xs xr : Ax0, x0, () 1 where x x x 1 n,..., a 1n vetor whh nludng the deon varable, atrx T A a and vetor b b1, b2,..., b are repetvely a atrx and a vetor nludng the rp real nuber. Alo, 1, 2,..., n a row vetor of rando nterval data. Sne the obetve funton oeffent of Proble (1) are rando, o th proble not well-defned. A a reult, we annot optze t lar to deternt ae. In order to deal wth uh SLP proble, everal deon approahe have been appeared n the lterature. Here, we fou on the Expetaton Optzaton (EO) ethod and then preent an extended veron of th odel whh n

5 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 75 faou a the expeted value odel, and ung th approah replae the obetve funton paraeter by t ean value. Now, aue that oe or all of the oeffent of 1, 2,..., n n the SLP proble are rando, hene, the an target of Proble (1) n the expetaton odel repreented a: n n E x E x x 1 J1 where and,..., denote the ean of rando varable 1 n E. ean the expetaton. Hene, Proble (1) wll be tranfored nto the followng proble: Max E z ( x ) E x t.. xs ( 2) n where S x Ax b x 0 ean value of x. R :,, and Fuzzy Matheatal Prograng E x In th eton, onder FMP odel a follow: Max f x, 0 3 t.. g x, a, () x 0, I, where I 1,...,, and,,..., x x1 x2 x n T ndate the a real vetor nludng the deon varable, and rando vetor T 1, 2,..., n nludng the obetve oeffent. The row vetor a how the th row of A a, where A a real n denonal atrx of tehnal oeffent. And, funton f and where I poe ontnuou property up to the eond dervatve, g

6 76 (IJMS) Vol. 11, No. 1, Wnter that f and g C, 1,...,. Alo denote a fuzzy extenon of on R whh ued to opare the left and the rght de of fuzzy ontrant (Dubo & Prade, 1980). Wth regard to g x, a 0, I doe not ake a rp feable regon, o, n order to produe a deternt feable area, the dea doe not provde onfdene level at whh t derable that the orrepondng - th fuzzy ontrant hold. Therefore, n order to obvate thoe entoned retrton, we ntrodue the followng odel: Max E f x, t.. g x, a 0, ( ) x 0, I where, E f x, how the ean of, f x. Let denote the ean of, and hene, the obetve funton an be learly wrtten a: E f x, f x, E f x,, So, Model (2) an be equvalently tranfored to: Max f x, 0, (5) t.. g x, a I, x 0, In order for a gnfant eleton of the eberhp funton for eah fuzzy ontrant, t refer to, f, 0, g x a thu, th ontrant wholly atfed, f g x, a p, o that the paraeter p the predefned axu tolerane fro zero, whh deterned by an expert deon aker, therefore, the -th ontrant ertanly volated. Note that, for g x, a 0, p, the eberhp funton onotonally dereang. Furtherore, when eberhp

7 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 77 funton of the ontrant are ondered n the lnear for, we have: 1, g x, a 0, g x, a Ax, b 1, 0 g x, a p, 1, 2, 3, ( 6) p 0, g x, a p I where 1,...,. Now, let u to begn wth the onept of feable oluton to the fuzzy prograng proble wth tohat paraeter of Model (3). The followng defnton prepared for th a. Defnton 1. Let 1,..., 01, be a vetor, and n X x R x 0, g x, a 0, I 1,...,. Then, the vetor x X naed an feable oluton of Model (3). Followng propoton enable u to defne feable et of Model (3) a an ntereton of all ut orrepondng to fuzzy ontrant. Propoton 1. Let 1,..., 01,, then X X, where n X x R x 0, g x, a 0, For I 1,..., (Naely, X the -utof the -th ontrant). Proof. For 1,..., 01,, let x X. Therefore, g x, a 0 n and fro X x R x, g x, a 0 0, we have x X, I. Therefore, x. Moreover, f x, we X 1 1 X 1 have x X I, 0 g x a and hene, x X. Therefore, the proof opleted.,, o

8 78 (IJMS) Vol. 11, No. 1, Wnter 2018 Propoton 2. Let,..., and 1,...,, where 1 for all. Then feablty of x ple the feablty of t. Proof. By the ue of defnton of ut and alo feablty of the oluton the proof traghtforward. n 01, let a oluton x R be uual feable to Proble () (a oluton n whh ha the ae atfaton degree n g x, a 0, or For a gven, the entoned ontrant). It ean that x, for all I X If,...,, 01, then x, whh ple that the feablty of Proble (3) an be undertood a a peal ae of the feablty. Therefore, we edately have the next reult. Reark 1. If Proble (3) feable, we learly onlude that X not epty. Defnton 2. Let be a fuzzy extenon of oon relaton and alo a oluton,..., T n x x1 x n (3), where 1,..., 01, and let, obetve n the for of axzaton. Therefore, x x x where X R be feable to Proble f x be a tohat,...,, 1 n n x R an effent oluton to Proble (3), f there X, E f x E f x, o that. no x Slarly, an effent oluton for the for of nzaton an be defned. It lear that any effent oluton to the entoned FMP ndeed a feable oluton to the FMP wth oe addtonal properte. Now, we gve the followng theore whh onerned to both portant ondton (that, neeary and alo uffent) for an effent oluton to Proble (3). We wll ee that th theore ha an portant role n the gven theoretal duon n our tudy.

9 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 79 Theore 1. Let 1,..., 01, where x 0, J = 1,2,...,n and alo x x 1 x n,...,, be a feable oluton to Proble n (3). Then, x R an effent optal oluton to Proble (3), where the obetve funton aued n the type of axzaton, f and only f the deon akng vetor x an optal oluton to the followng progra: Max f x, 1.. t g x, a p, I 1,...,, (7) x 0, J, where p the predefned axu tolerane. Proof. Let,...,, 1 01 and let x x 1 n, uh that x 0, J, be an effent oluton to Proble (3). Attar and Naer (201) by ung Defnton 1 and Equaton (6) onluded that x feable to Model (7), beaue g x a 0 equvalently g x, a 1 p for I Defnton 2, there no x X uh that E f x E f x, or. Alo, aordng to,,, t ean that x optal to Model (7). Converely, f x an optal oluton to Model (7), obvouly, x an feable oluton to Model (3) and hene, the optalty of x ple that the effeny of x. In Theore 1, we have dued a ethod to fuzzy atheatal prograng proble to obtan an effent oluton. If the reultng Proble (7) ha only one optal oluton, then we have proved that th oluton an effent oluton to the gven proble. In the ae of whh Proble (7) ha oe ultple optal oluton, n order to aheve a axu effent oluton, that an effent oluton wth, 1,...,, we perfor the followng two-phae approah. Note that the urrent uggeted

10 80 (IJMS) Vol. 11, No. 1, Wnter 2018 approah dfferent fro the laal two-phae ethod whh oon for olvng lnear prograng. In fat, n the propoed twophae approah, Equaton (7) an be olved n the frt phae, whle n the eond phae; a oluton obtaned whh ha hgher atfaton degree than the prevou oluton. Therefore, we obtan a ore ofortable agnent of the avalable reoure by ung th approah. Moreover, the aheved oluton by th ethod alo an effent oluton for the entoned proble. 0 0,..., 0 1 Let u all Proble (7) a phae 1 proble. Let and x, E f x, be the optal oluton of phae 1 wth 0 degree of effeny. 0 g x, a 0 Set, I. In Phae 2, we olve the followng proble Max 1 f x, f x,, t.. g x, a 1, x 0, 1 p, (8) where I and J. Let x, 1,..., be an optal oluton to Proble (8) (Phae 2). Then, the next portant reult at hand. Clearly, th reult an help u to undertand the valuable relaton between Proble (3) and (8). Theore 2. The optal oluton x to Proble (8) a axu effent oluton to Proble (3). Proof. Ung Proble (8), Propoton 2 and due to 0, t reult that x an 0 feable oluton to Proble (3) and th how that t feable n Model (5). Wth optalty of x n Model

11 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 81 (5) and oreover, f x, S f x, S of x n Proble (5) and f x, S f x, S, we wll have optalty. Thu, x an 0 effent oluton for Proble (3). Alo, due to the optalty, 1,..., n Proble (8), we have x and the potvty of the obetve funton oeffent g x, a 0, I. Now, aue that x not a axu 0 effent oluton for Proble (3). 0 Thu, there an effent oluton x for Proble (3), o that, I and for oe k, k k. where, g x, a 0, I and alo f x, S f x, S Thu, x, 1,..., feable to Proble (8) and k k 1 1, k 1, k 1 The proof opleted now. Next eton prepared to derbe our uggeted approah. Nueral Duon Now, we are at the plae that we would lke to explore the olvng proe for the extended odel by oe llutratve exaple. Exaple 1: Aue that the followng odel gven to olve: Max x t.. 5x1x 2 3x 3 x 16, 7x1x 2 3x 3 x 70, ( 9) 2x1x 2 9x 312x 90, x 0, 1,...,, where, 1,...,, 1, 2, 3, rando varable whh defned on oe probablty pae, F, P. Table 1 derbe the value of rando varable oeffent n obetve funton.

12 82 (IJMS) Vol. 11, No. 1, Wnter Table 1. Value of the Rando Varable Coeffent P The expetaton odel wth fuzzy ontrant forulated a: Max t.. 5x1x 2 3x 3x 16, 7x1x 2 3x 3 x 70, ( 10) 2x1x 2 9x 312x 90, x 0, 1,...,. Baed on the onept of the expetaton, fro the gven data fro Table 1, we have: 3, , 8, 12, o, the urrent progra an be re-wrtten a: Max 3x 15x 2 8x 312x t.. 5x1x 2 3x 3x 16, 7x1x 2 3x 3x 70, ( 11) 2x1x 2 9x 312x 90, x 0, 1,...,. wth the eberhp funton alo defned n Model (6) a followng:

13 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 83 1, Ax, b, p b Ax Ax, b Ax b, b p, 123,,, p 0, Ax b p, uh that the eleent of P p p p 1, 2, ,, repetvely the predefned axu tolerane fro b, 123,, (Fao, 1983). By Theore 1, we an rewrte Model (11) a follow: Max 3x15x 2 8x 3 12x t.. 5x1x 2 3x 3 x 16511, 7x1x 2 3x 3 x , ( 12) 2x1x 2 9x 3 12x , 0 1, 1, 2, 3, x,..., x 0. 1 Soe effent oluton wth atfaton degree whh deon aker' dere an be found n Table 2. Table 2. Soe Optal Soluton to Model (11) wth Dfferent Satfaton a b d e f a (0.5,0.5,0.2) (0.5,0.5,0.8) (0.5,0.1,0.5) (0.5,0.9,0.5) (0.5,0.5,0.5) T x x x x x If all of the atfaton degree are equal, then the feablty and effeny redue to la feablty and optalty (ee Table 1, olun f ). T Let x be an ,.,. effent oluton wth x a an optal obetve value (ee Table 1, olune ). In Phae 2, t

14 8 (IJMS) Vol. 11, No. 1, Wnter 2018 neeary to olve a ult-paraetr LP progra whh hown below: Max x15x 2 8x3 12x t.. 5x x 3x x 165 1, x x 3x x 700 1, ( 13) 2x1x 2 9x3 12x , , , , x,..., x 0. By olvng the above ult-paraetr progra, an optal oluton wll be aheved a: x 0., ,., Alo, T T x x We have Ax b Ax b 1 1, 1 3 3, and 2 Ax 2, b2 1 Thu, olvng the odel by the entoned approah, we an aheve a new optal oluton for Proble (9), whh not only t ha the optal value for the gven obetve, but alo t gve u a hgher eberhp value n. 2 Exaple 2: A anufaturng opany produe three produt n three proee. The ahnng te of eah produt n eah proe gven n Table 3. The axu avalable te of the proee per week for Proe I approxately 600 nute, for Proe II approxately 00 nute, and for Proe III approxately 200 nute. The proft of eah unt of produt A, B, and C repetvely 1, 2 and 3, whh are rando varable wth the expetaton 30, 12, and 11.

15 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 85 Table 3. The Mahnng Te Proe Produt A Produt B Produt C I II 5 7 III 3 2 To axze proft, the nuber of produt produed per week obtaned. Suppoe that the aount of produt A, B. and C wll be repetvely produed by x 1, x 2 and x 3 varable. Then, by ung the above value, the entoned progra forulated a: Max t.. 30x 112x 211x 3 9x13x25x3 600, 5x1x 2 7x3 00, 1 3x12x 2x3 200, x, x, x Suppoe that predefned axu tolerane fro b, 1, 2, 3are deterned by anager of the opany a p 1 60,p 2 0, and p3 20, repetvely. Now, by ung the eberhp funton whh defned n Model (6), we an rewrte Proble (1) a follow: Max 30x 112x 2 11x 3.t. 9x13x 2 5x , 5x1x 2 7x , 15 3x12x 2 x , 0 1, 1, 2, 3, x, x, x So, for 05., 12,, 3, th proble an be olved by ung oftware of LINGO 1.0 and an optal oluton obtaned a follow: x 7000,,

16 86 (IJMS) Vol. 11, No. 1, Wnter 2018 wth the optal obetve value T x 2100 Now, n order to obtan a axu effent oluton for the above entoned proble, t neeary to olve the followng ultparaetr LP proble, Max x112x 2 11x ,.t. 9x13x 2 5x , 5x1x 2 7x , 16 3x12x 2 x , , , , x 0, 1, 2, 3. By olvng the above paraetr lnear odel, we aheve the optal oluton below: x 7000,, Alo, T x T x Whle we have: Ax b Ax b Ax b, 05.,, 1,, The above reult onlude that the bet value for the atfaton degree of the obtaned optal oluton aheved. We ee that n the eond reoure, we ay need to redue the tolerane to 00. Conluon In th reearh, we frt extended the oon FLP to the tohat envronent. In partular, we ntrodued the novel approah to olve lnear prograng odel wth tohat paraeter and flexble ontrant. Alo, we preented the onept of feablty and effeny, where a vetor of atfaton degree whh are deterned by the deon aker. Thee onept help u to obtan ore flexble oluton to FMP proble. In addton, n the ae of

17 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 87 lnear proble, we have proved that the dered oluton an be aheved by olvng a orrepondng ult-paraetr LP. Aknowledgeent. Author would lke to greatly appreate the helpful oent of the anonyou referee to lead u for provng the earler veron of th anurpt.

18 88 (IJMS) Vol. 11, No. 1, Wnter 2018 Referene Attar, H., & Naer, S. H. (201). new onept of feablty and effeny of oluton n fuzzy atheatal prograng proble. Fuzzy Inforaton and Engneerng, 6(2), Ben Abdelazz, F., & Mar, H. (2005). Stohat prograng wth fuzzy lnear partal nforaton on probablty dtrbuton. European Journal of Operatonal Reearh, 162(3), Cadena, J. M., & Verdegay, J. L. (2000). Ung rankng funton n ult-obetve fuzzy lnear prograng. Fuzzy Set and Syte, 111(1), Caella, G., & Berger, R. L. (2001). Stattal nferene (2nd ed.). Paf Grove: Duxbury Pre. Dubo, D., & Prade, H. (1980). Rankng fuzzy nuber n the ettng of poblty theory.inforaton Sene, 30, Fao, A. V. (1983). Introduton to entvty and tablty analy n nonlnear prograng. New York, NY: Aade Pre. Grett, G. R., & Strzaker, D. R. (2001). Probablty and rando proee. Oxford Unverty Pre. Hop, N. V. (2007a). Fuzzy tohat goal prograng Proble. European Journal of Operatonal Reearh, 176(1), Hop, N. V. (2007b). Solvng fuzzy (tohat) lnear prograng proble ung uperorty and nferorty eaure. Inforaton Sene, 177, Hop, N. V. (2007). Solvng lnear prograng proble under fuzzne and randone envronent ung attanent value. Inforaton Sene, 177, Ikander, M. G. (2003). Ung dfferent donane rtera n tohat fuzzy lnear ult-obetve prograng: A ae of fuzzy weghted obetve funton. Matheatal and Coputer Modelng, 37(1-2), Ikander, M. G. (200a). A fuzzy weghted addtve approah for tohat fuzzy goal prograng. Appled Matheat and Coputaton, 15(2), Ikander, M. G. (200b). A poblty prograng approah for

19 Aeloraton of Verdegay Approah for Fuzzy Lnear Progra 89 tohat fuzzy ult-obetve lnear fratonal progra, Coputer and Matheat wth Applaton, 8(10), Ikander, M. G. (2005). A uggeted approah for poblty and neety donane nde n tohat fuzzy lnear prograng. Appled Matheat Letter. 18(), Luhandula, M. K. (2006). Fuzzy tohat lnear prograng: Survey and future reearh dreton. European Journal of Operatonal Reearh, 17(3), Mohan, C., & Nguyen, H. T. (2001). An nteratve atfng ethod for olvng ult-obetve xed fuzzy-tohat prograng proble. Fuzzy Set and Syte, 117(1), Naer, S. H., Ardl, E., Yazdan A., & Zaefaran, R. (2005). Splex ethod for olvng lnear prograng proble wth fuzzy nuber. World Aadey of Sene, Engneerng and Tehnology10, Naer, S. H., & Bavand, S. (2017). A uggeted approah for tohat nterval-valued lnear fratonal prograng proble. Internatonal Journal of Appled Operatonal Reearh, 13, Roelfanger, H. (2007). A general onept for olvng lnear ultrtera prograng proble wth rp, fuzzy or tohat value. Fuzzy Set and Syte, 158(17), Verdegay, J. L. (1982). Fuzzy atheatal prograng. Fuzzy Inforaton and Deon Proee, 231, 237.